Inhalt: The course provides an introduction to stochastic processes with applications in physics and chemistry. We will develop the mathematical theory in the language of modern probability theory, mostly following the book by Øksendal. The results are then applied to diffusion problems and non-Markovian transport (as in heterogeneous fluids), with the Brownian motion of a colloidal particle serving as paradigmatic example. Further, we will discuss the modelling of chemical reactions, the derivation of stochastic equations of motion from a many-particle Liouville dynamics, and recent developments in non-equilibrium physics. The course is given in English and accompanied by a mandatory problem class.

1. Stochastic processes and correlation functions
   (elements of probability theory, expectation, stochastic processes, dynamic correlation functions)
2. Molecular transport phenomena
   (Brownian motion of a tracer, complex fluids, fluctuation–dissipation theorem, correlations in space and time)
3. Stochastic differential equations
   (Ito and Stratonovich integral, martingale processes, Ito diffusion, Dynkin's formula)
4. Markov processes
   (Markov chains, diffusion processes, Fokker–Planck equation, detailed balance, first-passage time problems)
5. Statistical mechanics
   (linear response theory, Zwanzig–Mori projection formalism, exact Langevin equations)
6. Non-equilibrium thermodynamics
   (modern fluctuation theorems, stochastic thermodynamics, balance equations, Onsager relations)

Zielgruppe: M.Sc. Mathematik, M.Sc. Physics

Voraussetzungen: Analysis I - III

Literatur (alphabetisch):

• Dynkin: Markov processes (Springer, 1965)
• Gardiner: Handbook of Stochastic Methods (Springer, 2004)
• Hansen, McDonald: Theory of Simple Liquids (Academic, 2006)
• Kubo, Toda, Hashitsume: Statistical Physics II (Springer, 1998)
• Øksendal: Stochastic Differential Equations (Springer, 2010)
• Paul, Baschnagel: Stochastic Processes (Springer, 2013)
• van Kampen: Stochastic Processes in Physics and Chemistry (Elsevier, 2007)